## An Alternative Proof of Minimum Trace Reconciliation

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### Introduction
- Paper: An Alternative Proof of Minimum Trace Reconciliation, Prepared by Sakai Ando and Futoshi Narita, Authorized for distribution by Prachi Mishra, July 2022, WP/22/136.
- Key claim: Provides a simpler, more direct proof of minimum trace reconciliation (Wickramasuriya et. al. (2019)) by solving the first-order condition in the space of non-square matrices; shows this approach extends naturally to minimum weighted trace reconciliation (Panagiotelis et. al. (2021)).
- Motivation: The original proof in Wickramasuriya et. al. (2019) is indirect (bounds + constrained minimization of a lower bound) and not easy to extend.

### Setup
- Variables and dimensions:
  - y_t: m×1 vector, b_t: n×1 vector with m>n>0.
  - Linear constraint: y_t = S b_t (equation (1)).
  - S is m×n with last n rows equal to I_n: S = [ C; I_n ] (equation (2)), so S is full column rank for any C.
  - Feasible set: A = { y ∈ ℝ^m : [ I_{m−n} −C ] y = 0 } (equation (3)).
- Base forecast:
  - Base h-step ahead forecast ŷ_T(h) is unbiased: E_T[y_{T+h}] = E_T[ŷ_T(h)] (equation (4)).
  - Reconciled forecast defined via an n×m matrix P: ỹ_T(h) = S P ŷ_T(h) (equation (5)).
  - Unbiasedness constraint on P: P S = I_n (equation (6)).
- Forecast error covariance:
  - Let Σ = E_T[(y_{T+h} − ŷ_T(h))(y_{T+h} − ŷ_T(h))′], assumed invertible.
  - Reconciled forecast error covariance: E_T[(y_{T+h} − ỹ_T(h))(y_{T+h} − ỹ_T(h))′] = S P Σ P′ S′ (equation (7)).
- Objective (minimum-trace reconciliation):
  - Wickramasuriya et. al. (2019) prove that the P minimizing the trace subject to P S = I_n is
    (S′ S^{-1} S)^{-1} S′ S^{-1} = argmin_{P∈ℝ^{n×m}} tr[S P S Σ P′ S′]  s.t.  P S = I_n.  (equation (9) as stated in the source)

### Proof of (9) in Wickramasuriya et. al. (2019) (summarized)
- Two-step approach in the original proof:
  1. Bound the objective from below using Weyl’s inequalities: tr[S P S Σ P′ S′] ≥ tr[P S Σ P′], ∀P ∈ ℝ^{n×m} (equation (10)).
  2. Solve the minimization problem of the lower bound subject to P S = I_n:
     (S′ S^{-1} S)^{-1} S′ S^{-1} = argmin_{P∈ℝ^{n×m}} tr[P S Σ P′]  s.t.  P S = I_n. (equation (11))
- Gap: need to show minimizers of the original objective and the lower-bound objective coincide.

### Alternative proof (first-order / Gateaux derivative approach)
- Function space and inner product:
  - Consider (ℝ^{n×m}, ⟨ , ⟩) with Frobenius inner product ⟨A,B⟩ = tr(A′ B) (equation (12)).
- Lagrangian and stationarity:
  - L(P) = tr(S P S Σ P′ S′) + tr[Λ′(I_n − P S)] with Λ an n×m Lagrange multiplier (equation (13)).
  - Gateaux differential set to zero for all H ∈ ℝ^{n×m} yields (equation (14))
    lim_{α→0} (L(P+αH) − L(P))/α = tr( H ( 2 S P′ S′ Σ S − S Λ′ S ) ) = 0.
  - Since this holds for all H, obtain first-order condition
    2 S P′ S′ Σ S = S Λ′ S ⇒ P′ S′ Σ S = (1/2) S^{-1} S Λ′ S. (equation (15))
  - Multiply by S′ on the left and use S P S = S to obtain
    (S′ S Σ^{-1} S)^{-1} S′ Σ S = (1/2) Λ′ S (equation (16) in source formatting).
- Conclusion:
  - Rearranging yields the closed-form P:
    P′ S′ Σ S = S^{-1} S (S′ S Σ^{-1} S)^{-1} S′ Σ S ⇒ P = (S′ S Σ^{-1} S)^{-1} S′ Σ^{-1}. (equation (17) as presented in the source)
  - QED: the first-order condition is sufficient because the objective is quadratic and the constraint is linear.

### Extension: Minimum weighted trace reconciliation (Panagiotelis et. al. (2021))
- Motivating example: GDP expenditure identity
  - Ŷ = Ĉ + Î + Ĝ + X̂  (equation (18) as given).
- Weighted trace objective (diagonal weights example):
  - Minimize weighted sum of variances with positive weights ω_i, ∑ ω_i V[forecast errors] (equation (20)).
  - Matrix form: min_P tr( ω S P Σ P′ S′ )  s.t.  P S = I_n, where ω is m×m diagonal with (i,i) element ω_i (equation (21)).
- Claim (from source): For any symmetric and invertible m×m matrix ω, the solution to (21) is
  P = (S′ S Σ^{-1} S)^{-1} S′ Σ^{-1}. (equation (22) as stated)
- Proof sketch (parallel to unweighted case):
  - Lagrangian: L(P) = tr( ω S P Σ P′ S′ ) + tr[ Λ′( S − S P S ) ] (equation (23) in source).
  - Gateaux derivative leads to condition (equation (24) → (25) in source)
    2 S P′ S′ ω S = S Λ′ S ⇒ P′ S′ ω S = (1/2) S^{-1} S Λ′ S.
  - Premultiplying by S′ and using S P S = S yields
    (S′ ω S) = (1/2) S′ S Σ^{-1} S Λ′ S ⇒ (S′ S Σ^{-1} S)^{-1} S′ ω S = (1/2) Λ′ S (equation (26)).
  - Using invertibility of S′ ω S, deduce P = (S′ S Σ^{-1} S)^{-1} S′ Σ^{-1} (equation (27)).
- Insight:
  - The optimal P is independent of ω as long as ω is symmetric and invertible.
  - Interpretation: no trade-off between variables — the choice matrix P has sufficient degrees of freedom to minimize each variable’s forecast-error variance individually.
- Further observation:
  - Using Takagi’s factorization ω = A′ A (A full-rank square), the objective becomes tr( A S P Σ P′ S′ A′ ) (equation (28)).
  - Allowing A to be a full-rank non-square matrix (e.g., A = (S′ Σ^{-1} S)^{-1} S′) can collapse the objective to tr( P S Σ P′ ) (equation (29)), providing a route to understand why the lower-bound minimization in Wickramasuriya et. al. (2019) attains the same formula.

### Conclusion
- The paper provides an alternative, direct proof of minimum trace reconciliation by solving first-order conditions in matrix space.
- The same approach extends to minimum weighted trace reconciliation (Panagiotelis et. al. (2021)), unifying trace and weighted-trace results and offering pedagogical and conceptual clarity on why the weighting matrix does not affect the optimal reconciler under symmetry and invertibility conditions.

*An Alternative Proof of Minimum Trace Reconciliation, IMF Working Paper WP/22/136.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022136-print-pdf.pdf_